Zeno of Elea & the Evolution of Infinity Kornilowicz, Gabriel Chu, Dan
Zeno and his Paradoxes Born in 490 BCE in Elea, Italy Student of the Eleatic philosopher Parmenides Upon his arrival in Athens with his teacher he introduced his book of paradoxes, of which very little remains Source: https://probaway.wordpress.com/category/philosophers-squared/
Parmenides Born 515 BCE in Elea, Italy Greek philosopher who focused on Monism The belief that there exists only one thing in the universe Source: http://www.cplong.org/digitaldialogue/digital_dialogue_17_parmenides/
Zeno s Goals There is some contention around the goals of zeno and his paradoxes: There is the belief that he was merely trying to prove his teachers Monistic arguments There is also the belief that instead of directly proving his teacher, his paradoxes existed to show how ridiculous the oppositions beliefs were, thus indirectly defending his teacher.
The Arrow Time is made up of an infinite number of points An object must occupy a space equal to itself at all times So, an arrow in flight at any and all moments is not moving y 1 1 speed=0 y 1 2 2 speed=0 y 2 t1 t2 t
The Arrow We know that in calculus the derivative of the position of an object with respect to time is the velocity: y 1 1 y velocity >0 1 2 2 y velocity >0 2 t1 t2 t
Dichotomy (The Racetrack) Before a runner can reach the end, they must first travel half Before a runner can reach the halfway, they must first travel onefourth Before a runner can reach the onefourth, they must first travel one-eighth Therefore the runner must travel an infinite distance Source: https://1badnavajo.files.wordpress.com/2013/04/track.jpg
Dichotomy (The Racetrack) n=1 Zeno believed this to converge to infinity Actually converges to one Lack of calculus 1 1/2 1/4 1/8 1/16 1/32...
Achilles and the Turtle Imagine that Achilles and a turtle are having a race Achilles is much faster than the turtle The turtle receives a head start When Achilles begins running, he must first catch up to where the turtle once was By the time he does this, the turtle has moved on He must then catch up with the turtle once more But by this time, the turtle has moved again Etc Zeno assumes that space and time are infinitely divisible
Achilles and the Turtle Source: http://www.iep.utm.edu/wp-content/media/achilles_tortoise.jpg Straw man argument Distance Achilles's has to travel is not actually infinite Imagine Achilles travels: d1 = catch up to p1 d2 = catch up to p2 d3 = catch up to p3 d1 + d2 + d3 +!= Will eventually converge to a constant
Infinity Post - Zeno Zeno s paradoxes caused Aristotle to redefine the concept of infinity Producing the concepts of Actual Infinity Potential Infinity Actual Infinity: Aristotle described it as a set with a beginning and end that at this moment contains an infinite number of terms, he did not believe this could be achieved in nature Example: N Potential Infinity: Aristotle believed it to be a process that will continuously operate for an infinite amount of time Example: Splitting a group in two continuously
Aristotle on Zeno s Paradoxes Aristotle believed that Zeno was incorrect in saying, for example with the dichotomy, a person must walk through an actually infinite number of points He said that you instead traverse a potentially infinite number of points We now know that Zeno s fault did not lie with his interpretation of actual infinity, in fact there are an actually infinite number of points that need to be traversed. He was incorrect in thinking that one cannot traverse this.
Infinity Today Potential Infinity: Actual Infinity: Cat paradox
Infinity Today Computer Science Growth functions Both n^3 and n^2 will reach infinity, but which one does it faster? Sometimes used as some large number Source: http://i50.tinypic.com/f23nuh.jpg
Works Cited Dowden, Bradley. "Zeno s Paradoxes." Internet Encyclopedia of Philosophy. University of Tennessee Martin, n. d. Web. 01 Mar. 2015. Huggett, Nick. "Zeno's Paradoxes." Stanford University. Stanford University, 30 Apr. 2002. Web. 01 Mar. 2015. Palmer, John, "Zeno of Elea", The Stanford Encyclopedia of Philosophy (Spring 2012 Edition), Edward N. Zalta (ed.) Palmer, John, "Parmenides", The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta (ed.)